iti 
equation of Lagrange, which ‘8 be written thus: 
mn 
da 
=.m (Sze + Tay eo 5 82) = = 3>.— 
where 7, by (2), (4), (5), is equal to the known expression, 
r= v{(e— 2+ y¥—y? + @ — 2). 
If the law of the attraction were supposed different from 
that of the inverse square, a different function of 7, instead of 
r—', should be multiplied by the product of two masses. 
But further, it is not difficult so to operate on the formula 
(3), as to deduce from it another equation which shall be equi- 
valent to the forms that were proposed by the present author, 
in his papers ** On a General Method in Dynamics” (pub- 
lished in the Philosophical Transactions*), as being, at least 
theoretically, forms for the integrals of the differential equa- 
tions of motion of any system of attracting bodies. For if we 
observe, that by the principles of the calculus of variations, 
combined with those of the method of vectors, we have the 
identity, 
da d?a + d’ada = d(dada + dada) — d(da’); 
and if we write é 
eV 3-(G)$: ©) 
denoting by v the magnitude or degree (but not the direction) 
of the velocity of the body of which the vector is a; we may 
transform the ae (3) into the foNowing : 
ds 
dé 
which, when operated upon by the characteristic { dé, that 
. : 
mv mm 
2 (sat +— fe 34) 4 3(3™e 4 3) =o; (7) 
is, when integrated once with respect to the time from 0 to ¢, 
~ becomes 
2.7 (3e a a 7a — e544) + or=0, (8) 
* 1834, Part II. 1835, Part I. 
e 2 
